Search results for "18B40"

showing 4 items of 4 documents

The identity type weak factorisation system

2008

We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.

Class (set theory)Pure mathematicsGeneral Computer ScienceDependent type theoryHomotopiaType (model theory)Identity (music)Theoretical Computer Science510 - Consideracions fonamentals i generals de les matemàtiquesCombinatorics18C50Mathematics::Category TheoryFOS: MathematicsCategory Theory (math.CT)Univalent foundationsAxiomMathematicsHomotopy03B15; 18C50; 18B40Mathematics - Category TheoryIdentity type weak factorisation systemMathematics - LogicTipus Teoria dels03B15Type theory18B40Homotopy type theoryLogic (math.LO)Weak factorisation systemIdentity typeComputer Science(all)
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Butterflies in a Semi-Abelian Context

2011

It is known that monoidal functors between internal groupoids in the category Grp of groups constitute the bicategory of fractions of the 2-category Grpd(Grp) of internal groupoids, internal functors and internal natural transformations in Grp, with respect to weak equivalences (that is, internal functors which are internally fully faithful and essentially surjective on objects). Monoidal functors can be equivalently described by a kind of weak morphisms introduced by B. Noohi under the name of butterflies. In order to internalize monoidal functors in a wide context, we introduce the notion of internal butterflies between internal crossed modules in a semi-abelian category C, and we show th…

Discrete mathematicsPure mathematicsButterflyFunctorInternal groupoidWeak equivalenceGeneral MathematicsSemi-abelian categoryFunctor categoryContext (language use)Mathematics - Category TheoryBicategory of fractionBicategoryMathematics::Algebraic TopologyWeak equivalence18D05 18B40 18E10 18A40Surjective functionMorphismMathematics::Category TheoryFOS: MathematicsCategory Theory (math.CT)Abelian groupMathematics
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The snail lemma for internal groupoids

2019

Abstract We establish a generalized form both of the Gabriel-Zisman exact sequence associated with a pointed functor between pointed groupoids, and of the Brown exact sequence associated with a fibration of pointed groupoids. Our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive coequalizers.

Pure mathematicsExact sequenceLemma (mathematics)Internal groupoid Snail lemma Fibration Snake lemmaAlgebra and Number TheoryFunctorMathematics::Operator Algebras010102 general mathematicsFibrationMathematics - Category Theory01 natural sciences18B40 18D35 18G50Settore MAT/02 - AlgebraMathematics::K-Theory and HomologyMathematics::Category Theory0103 physical sciencesFOS: MathematicsCategory Theory (math.CT)Regular category010307 mathematical physics0101 mathematicsMathematics::Symplectic GeometryMathematics
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Bipullbacks of fractions and the snail lemma

2017

Abstract We establish conditions giving the existence of bipullbacks in bicategories of fractions. We apply our results to construct a π 0 - π 1 exact sequence associated with a fractor between groupoids internal to a pointed exact category.

Pure mathematicsLemma (mathematics)Exact sequenceInternal groupoidAlgebra and Number Theory010102 general mathematicsMathematics - Category TheoryBicategory of fraction18B40 18D05 18E35 18G5001 natural sciencesMathematics::Algebraic TopologySettore MAT/02 - AlgebraExact categoryMathematics::K-Theory and HomologyMathematics::Category Theory0103 physical sciencesFOS: MathematicsBipullbackSnail lemmaCategory Theory (math.CT)010307 mathematical physics0101 mathematicsMathematics
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